The chat windows in the overlay and on the regular desktop seem to be completely separate. When I start chatting in the overlay and minimize the game (on Xfce), there is no chat window open as a "desktop window" replicating the in-game chat. I believe this is different from (and worse than on) Windows because using Steam with Wine causes the chat window to appear not only as part of the overlay but also on the desktop when someone sends me an invite or a chat message (not sure about what happens when I am the one initiating the conversation).